Do not use a calculator in answering this question. Showing your working, find the complex numbers $z$ and $w$ which satisfy the simultaneous equations
$\begin{aligned} 2\text{i}z + w &= 5 + 7\text{i}, \\ (4 - \text{i})w^* &= z - 3\text{i}. \end{aligned}$

1. Rearrange the second equation to make $z$ the subject.

$z = (4 - \text{i})w^* + 3\text{i}$

2. Substitute $z$ into the first equation and simplify to obtain an equation in terms of $w$ and $w^*$.

$2\text{i}[(4 - \text{i})w^* + 3\text{i}] + w = 5 + 7\text{i}$
$(8\text{i} - 2\text{i}^2)w^* + 6\text{i}^2 + w = 5 + 7\text{i}$
$(2 + 8\text{i})w^* - 6 + w = 5 + 7\text{i}$
$w + (2 + 8\text{i})w^* = 11 + 7\text{i}$

3. Let $w = x + y\text{i}$, substitute $w$ and $w^*$ into the equation, and expand.

Let $w = x + y\text{i}$ and $w^* = x - y\text{i}$, where $x, y \in \mathbb{R}$.
$(x + y\text{i}) + (2 + 8\text{i})(x - y\text{i}) = 11 + 7\text{i}$
$x + y\text{i} + 2x - 2y\text{i} + 8x\text{i} - 8y\text{i}^2 = 11 + 7\text{i}$
$x + y\text{i} + 2x - 2y\text{i} + 8x\text{i} + 8y = 11 + 7\text{i}$
$(3x + 8y) + (8x - y)\text{i} = 11 + 7\text{i}$

4. Equate the real and imaginary parts to form a pair of simultaneous equations.

Equating real parts:
$3x + 8y = 11 \quad \text{--- (1)}$

Equating imaginary parts:
$8x - y = 7 \quad \text{--- (2)}$

5. Solve the simultaneous equations to find $x$ and $y$, thus obtaining $w$.

From (2), $y = 8x - 7$.
Substitute into (1):
$3x + 8(8x - 7) = 11$
$3x + 64x - 56 = 11$
$67x = 67 \implies x = 1$
$y = 8(1) - 7 = 1$
Therefore, $w = 1 + \text{i}$.

6. Substitute $w^*$ back into the expression for $z$ to find its value.

Using $w^* = 1 - \text{i}$:
$z = (4 - \text{i})(1 - \text{i}) + 3\text{i}$
$z = 4 - 4\text{i} - \text{i} + \text{i}^2 + 3\text{i}$
$z = 4 - 5\text{i} - 1 + 3\text{i}$
$z = 3 - 2\text{i}$